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Lin introduced the partition function $\text{PDO}_t(n)$, which counts thetotal number of tagged parts over all the partitions of $n$ with designatedsummands in which all parts are odd. For $k\geq0$, Lin conjectured congruencesfor $\text{PDO}_t(8\cdot3^kn)$ and $\text{PDO}_t(12\cdot3^kn)$ modulo$3^{k+2}$. In this article, we develop a new approach to study thesecongruences. We study the generating functions of $\text{PDO}_t(8\cdot3^kn)$and $\text{PDO}_t(12\cdot3^kn)$ modulo $3^{k+3}$ for certain values of $k$. Wealso study $\text{PDO}_t(n)$ modulo powers of $2$. We establish infinitely manycongruences for $\text{PDO}_t(n)$ modulo $8$ and $32$. We prove severalcongruences modulo small powers of $2$ and discuss the existence of congruencesmodulo arbitrary powers of $2$ similar to those in Lin's conjecture. Inreference to this, we also pose some problems for future work.

Congruences for the partition function $\text{PDO}_t(n)$ modulo powers of $2$ and $3$